Questions tagged [lower-bounds]
The lower-bounds tag has no usage guidance.
137 questions
24
votes
4
answers
5k
views
Super-linear time complexity lower bounds for any natural problem in NP?
Do we know any problem in NP which has a super-linear time complexity lower bound? Ideally, we would like to show that 3SAT has super-polynomial lower bounds, but I guess we're far away from that. I'd ...
- 2,416
20
votes
1
answer
1k
views
Quantitative lower bounds related to Zhang's theorem on bounded gaps
Let $\mathcal{H}=\left\{ h_{i}\right\} _{i=1}^{k}$ be an admissible set, and define $$\pi_{\mathcal{H}}(x)=\left|\left\{ n\leq x\ :\ \exists\ i,j\leq k,\ i\neq j\ \text{such that both }n+h_{i},\ n+h_{...
- 11.4k
17
votes
6
answers
10k
views
Lower bounds for chromatic number of a graph
I am trying to find a good lower bound for chromatic number of one family of graphs. I'm curious what are the known lower bounds for chromatic number. There are two obvious: $\chi(G) \geq \omega(G)$ ...
- 1,791
12
votes
0
answers
815
views
Lower bounds for linear forms of logarithms (a la Baker)?
Let $\lambda_1$, $\lambda_2$, and $a$ be three fixed complex algebraic
numbers.
For a given integer $n$, write
$\Theta(n) = \arg(a \lambda_1^n + \lambda_2^n)$.
Assuming $\Theta(n)$ is not zero, I am ...
- 121
11
votes
3
answers
1k
views
What is the limit of $a (n + 1) / a (n)$?
Let $a(n) = f(n,n)$ where $f(m,n) = 1$ if $m < 2 $ or $ n < 2$ and $f(m,n) = f(m-1,n-1) + f(m-1,n-2) + 2 f(m-2,n-1)$ otherwise.
What is the limit of $a(n + 1) / a (n)$? $(2.71...)$
11
votes
2
answers
1k
views
Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold
It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the (...
- 3,223
11
votes
0
answers
536
views
Bounding the probability that a random variable is maximal
Question: Suppose we have $N$ independent random variables $X_1$, $\ldots$, $X_N$ with finite means $\mu_1 \leq \ldots \leq \mu_N$ and variances $\sigma_1^2$, $\ldots$, $\sigma_N^2$.
I am looking ...
- 119
9
votes
1
answer
788
views
The expected value of product of random variables which have the same distribution but are not independent
Given a positive integer $k$, is there a positive real number $c(k)$ such that $\mathbb{E}\left(\prod_{i=1}^k X_i\right)\geq c(k)$ for any $k$-random variables $X_1,X_2,\ldots,X_k$ which all have the ...
- 99
8
votes
2
answers
1k
views
Lower bound on exponential sums
Let $k\geq 2$. Consider the following norm of exponenetial sum:
$$
I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy.
$$
Bourgain mentioned on Page 118 of
https://...
- 751
8
votes
1
answer
605
views
lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?
Are there known any lower and upper bounds for
$$
\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k,
$$
where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$?
Or at least is it known ...
- 841
8
votes
2
answers
748
views
Lower bound on the entries of the Perron vector
Let $A$ be a matrix that satisfies all the conditions of Perron- Frobenius theorem. From the theorem it is known that the entries of the eigenvector corresponding to the largest eigenvalue will be ...
- 183
8
votes
0
answers
271
views
Restricted divisor summatory function
I have a problem that boils down to prove that the succession $\{a_n\}$ tends to infinity, where
$$a_n:=1+\sum_{0\leq j<n}D_{2j+1}(n-j)$$
and $D_{m}(n)$ is the number of divisors $d>1$ of $n$ ...
- 361
7
votes
1
answer
339
views
lower bound for Perron-Frobenius degree of a Perron number
A Perron number is an algebraic number which is greater than one in absolute value and is greater than all of its Galois conjugates in absolute value as well. Lind's theorem states that any Perron ...
- 858
7
votes
2
answers
3k
views
Lower bound for Euler's totient for almost all integers
Let $\varphi(n)$ be the Euler's totient function. It is well know that $\liminf_{n \to \infty} \frac{\varphi(n)}{n / \log \log n} = e^{-\gamma}$, so that for $\varepsilon > 0$ it results $\frac{\...
- 285
7
votes
1
answer
310
views
Is this lower bound for the size of minimal vertex cover new/interesting?
I have found this lower bound for the size of minimal vertex cover (and proved it).
If a simple connected graph G on n vertices has largest and smallest eigenvalues $\lambda_1,\lambda_n$, ...
- 133
6
votes
2
answers
1k
views
Nontrivial lower bound on the sum of matrix norms
Let $X, V\in\mathbb{R}^{n\times r}$ such that $X^\top V$ is symmetric. The central quantity I care about is
\begin{equation}
\|XV^\top\|_{F}^2+\|X^\top V\|_{F}^2 +[\text{Tr}(X^\top V)]^2.
\end{...
- 515
6
votes
2
answers
259
views
Distance of low-rank matrices to the identity for the $\infty$-norm
I am trying to get a lower bound (or even the exact value) of
$$
\min_{X \in \mathbb{R}^{n\times n}} \|X - I_n\|_{\infty} \enspace \text{s.t.} \enspace \mbox{Rank}(X) = m
$$
where $m \leq n$, and the ...
- 197
6
votes
2
answers
743
views
bound for zeros of a polynomial with bounded integer coefficients
Let $f$ be a monic polynomial with bounded integer coefficients and such that all zeros are (in absolute value) greater than $1$. How close can the zeros of $f$ reach $1$ (in absolute value)?
More ...
- 577
6
votes
2
answers
4k
views
tight bounds on probability of sum of laplace random variables.
Are there tight upper and lower bounds on the density of the sum of $n$ i.i.d laplace random variables that depend on $n$ and the individual laplacian densities?
- 111
6
votes
1
answer
1k
views
Simple lower bounds for Bell numbers (number of set partitions)?
The $n$-th Bell number $B_n$ represents the number of distinct partitions of a set with $n$ distinguished elements.
It can be expressed as the infinite sum $B_n = (1/e)\sum_{k=1}^{\infty} (k^n/k!)$, ...
- 2,350
5
votes
4
answers
917
views
Limit of a sum with binomial coefficients
Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$
$$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$
$$C_k = \frac{\sum_{i=1}^k(...
- 155
5
votes
3
answers
4k
views
Proofs of Lower Bounds for Ramsey Numbers?
As a sort of dual question to this question, I am wondering what proofs people know of lower bounds on Ramsey numbers $R(k, k)$. I know of two proofs: there is Erdos's beautiful probabilistic ...
- 23k
5
votes
1
answer
1k
views
Best lower bound for off-diagonal Ramsey numbers
What are the current best lower bounds for off-diagonal Ramsey numbers $R(k,l)$ with $l$ of order unity and asking for asymptotic behavior for large $k$, such as $R(k,4)$, $R(k,5)$, and so on? (...
5
votes
2
answers
670
views
Lower Bounds for the Roots of Polynomials
I'm interested in the "size" of the roots of a sequence of Taylor Polynomials of an entire function.
For example, consider $\mathrm f(z) = \mathrm e^z$. The Taylor Polynomials, or $k$-jets, are
$$\...
- 310
5
votes
3
answers
148
views
Fast computation of a ball with radius r with largest number of input points
We are given a set S of n points equipped with some metric and an integer $r>0$. We define $B(x,r) \subseteq S$ (the ball with radius r centered in x) to be the set of points in S within distance r ...
- 51
5
votes
1
answer
927
views
Hausdorff distance is a lower (or upper bound) for what probability metric?
In a metric space $X=(X, d)$, given a probability measure $\mu$ and two subsets $A$ and $B$ of positive measure, it's not hard to prove that
$$
d(A, B) \le W(\mu|_A, \mu|_B),
$$
where
$d(A, B):= \...
- 6,853
5
votes
1
answer
271
views
Feasibility of linear programs
It's known that finding the intersection of n halfplanes in 2-d takes $\Omega(n\log n)$ time. Does the lower bound apply if we change the question to deciding whether the intersection is non-empty?
- 201
5
votes
0
answers
84
views
Smaller root of a difference of products of polynomials with integer bounded coefficients
Is there a positive constant $K>0$ such that
for every polynomials $f_1,\dots,f_4 \in\mathbb{Z}[X]$ with coefficients in {-1,0,1}, every positive root $x$ of the polynomial
$$g=f_1f_2-f_3f_4$$
...
4
votes
2
answers
1k
views
Lower bound on the first eigenvalue of the Laplacian of a Riemann surface with constant negative scalar curvature
A friend in physics asked this question, and I didn't know the answer.
Are there lower bounds on the first eigenvalue of the Laplacian of a Riemann surface equipped with a metric of constant negative ...
- 5,974
4
votes
2
answers
275
views
An inequality involving binomial coefficients and the powers of two
I came across the following inequality, which should hold for any integer $k\geq 1$:
$$\sum_{j=0}^{k-1}\frac{(-1)^{j}2^{k-1-j}\binom{k}{j}(k-j)}{2k+1-j}\leq
\frac{1}{3}.$$
I have been struggling with ...
- 155
4
votes
1
answer
258
views
Bounds for the crossing number in terms of the braid index?
Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?
For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
- 9,114
4
votes
1
answer
168
views
Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound
Consider bivariate copulas $C_1$ and $C_2$ with $\max\{C_1(u,v), C_2(u,v)\}< M_2(u,v)$ for all $u,v \in(0,1)$, where $M_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound.
Is there a ...
- 143
4
votes
1
answer
749
views
Lower bound on the tail of the hypergeometric distribution
Suppose there is a bag with $M$ white marbles and $N - M$ black marbles. Let $H(n, N, M)$ be a random variable which is number of white marbles in a draw, without replacement, of $n$ marbles from a ...
- 2,221
4
votes
1
answer
441
views
A lower bound on the $L^2$ norm of a Dirichlet polynomial
The Question. Suppose $0 < \alpha < \beta$ are fixed, and $a_n$ is an arbitrary sequence of real numbers. Is it known how to bound from below
\begin{equation*}
\int_0^{T} \Big| \sum_{\alpha T &...
- 4,661
4
votes
1
answer
232
views
Expectation over Pareto Sums
Given $K$ iid random variables $x_i$ with uniform distribution on $(0,1]$
and a constant $\alpha > 0$, the random variable $x_i^{-\alpha/2}$ is Pareto-distributed with scale parameter $1$ and ...
- 41
4
votes
0
answers
131
views
Log of a truncated binomial
Let $X$ follow a binomial distribution with $n$ trials and success probability $p$, and let $0\leq k\leq n$. Are there any natural approximations or bounds for the ratio $$\frac{\boldsymbol{E}\log\...
- 4,049
3
votes
1
answer
860
views
Lower bounds on the error term of the prime number theorem
Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t.
$$f(x)\ll |\psi(x) - x|$$
where $\psi$ is the Chebyshev function.
- 1,890
3
votes
1
answer
345
views
Simple anticoncentration bound for binomially distributed variable
The following question, which arose during my research, seems deceivingly simple to me, but I could not find any elegant and formal argument.
For a binomially distributed variable $X \sim \text{Bin} \...
- 85
3
votes
3
answers
6k
views
Probability of a random variable greater than its expected value
We have a lot of probabilities lower bounding as (e.g. chernoff bound, reverse markov inequality, Paley–Zygmund inequality)
$$
P( X-E(X) > a) \geq c, a > 0 \quad and \quad P(X > (1-\theta)E[...
- 141
3
votes
2
answers
253
views
Given a polynomial constraint equation in $n$ variables, can one conclude that the sum of the variables is non-negative?
Currently I'm stuck as follows;
at least a positive proof if $n=3$ would be a great nice-to-have!
Consider real numbers $x_1,x_2,\dots,x_n$ satisfying
$$\prod^n_{k=1}\left(1-x_k^2\right)\:=\:\...
- 489
3
votes
1
answer
87
views
If $X \sim N(0,I_m)$, what is a necessary and sufficient condition on $u_m > 0$ such that $\lim\sup_{m\to \infty} P(\|X\|^2 \ge u_m|X_1|) = 1$
Let $m$ be a large positive integer and $X=(X_1,\ldots,X_m) \sim N(0,I_m)$. I wish to show that the squared norm of $X$ is much much bigger than the absolute value of any of the $X_j$'s. For example, ...
- 6,853
3
votes
2
answers
283
views
Lower bound for Euler's function
Euler function is defined, for $|x|\le 1$, as follows:
$$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$
Upper bounds for $\phi$ can be simply derived from ending the product early, e.g.
$$\phi(x)<\prod_{i=1}^...
- 618
3
votes
2
answers
233
views
Lower bounding decoding error in a noisy adversarial channel
Problem description
Suppose we have a finite alphabet $\mathcal{X}$, where each letter $X \in \mathcal{X}$ indexes into some fixed set of distributions, $\{P_{1},\ldots,P_{|\mathcal{X}|}\}$. For ...
- 181
3
votes
1
answer
167
views
Lower bound on the sum of pmf squared of a hypergeometric distribution
I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...
- 33
3
votes
2
answers
1k
views
Large deviation upper bound for Chi-squared random variable
Let $X \sim \chi^2_n$ random variable. I am looking for a large deviation upper bound for $X$. The answer here, says that
Since you said that you're looking for an upper bound, it should also be ...
- 131
3
votes
1
answer
199
views
Looking for tighter bounds on a certain solution of a nonlinear equation
I have to solve an equation which is $$\sum_{i=1}^N x_i = \sum_{i=1}^N y_i,$$ where
$$x_i = \frac{z_i}{1 + (K_i - 1) w}$$ and $$y_i = \frac{K_i z_i}{1 + (K_i - 1) w}.$$
The $z_i$ are all positive ...
- 1,440
3
votes
1
answer
575
views
Minimum number of perfect matchings in a regular bipartite graph
Is there a lower bound on the number of perfect matchings in a $k$-regular bipartite graph?
One can use Hall's marriage theorem and induction on $k$ to derive the lower bound of $k$. I can't come up ...
- 217
3
votes
1
answer
679
views
using polynomials as lower / upper bound?
I'm interested in the question of given a differentiable and bounded function $f(\vec{x})$ (over a single variable or multiple variables, over a bounded domain $D$), finding a pair of polynomials $p_1(...
- 195
3
votes
2
answers
190
views
Techniques for showing optimality of given packing
There are some natural packing problems that have been asked in mathematics. Some of them are:
1)How many balls can be placed with in a cube?
2)How many equidistant points can be place on the ...
- 13.9k
3
votes
1
answer
165
views
Non-asymptotic tail bounds for $D_{\text{Hellinger}}(P\|\hat{P}_N)$
Let P be a distribution on a finite set of size $k$ and let $\hat{P}_N=(N_1/N,\ldots,N_k/N)$ be the empirical distribution (frequencies) from a samples of size $N$. Consider the Hellinger distance ...
- 6,853